getdoc849d. 140KB Jun 04 2011 12:04:36 AM
Elect. Comm. in Probab. 14 (2009), 261–269
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
AN EASY PROOF OF THE ζ(2) LIMIT IN THE RANDOM ASSIGNMENT PROBLEM
JOHAN WÄSTLUND 1
Department of Mathematical Sciences,
Chalmers University of Technology,
S-412 96 Gothenburg, Sweden
email: [email protected]
Submitted April 4, 2008, accepted in final form December 3, 2008
AMS 2000 Subject classification: 60C05, 90C27, 90C35
Keywords: minimum matching, graph, exponential
Abstract
The edges of the complete bipartite graph Kn,n are given independent exponentially distributed
costs. Let Cn be the minimum total cost of a perfect matching. It was conjectured by M. Mézard
and G. Parisi in 1985, and proved by D. Aldous in 2000, that Cn converges in probability to π2 /6.
We give a short proof of this fact, consisting of a proof of the exact formula 1+1/4+1/9+· · ·+1/n2
for the expectation of Cn , and a O(1/n) bound on the variance.
1 Introduction
We consider the following random model of the assignment problem: The edges of an m by n
complete bipartite graph are assigned independent exponentially distributed costs. A k-assignment
is a set of k edges of which no two have a vertex in common. The cost of an assignment is the
sum of the costs of its edges. Equivalently, if the costs are represented by an m by n matrix, a
k-assignment is a set of k matrix entries, no two in the same row or column. We let Ck,m,n denote
the minimum cost of a k-assignment. We are primarily interested in the case k = m = n, where
we write Cn = Cn,n,n .
The distribution
of Cn has been investigated for several decades. In 1979, D. Walkup [27] showed
that E Cn is bounded as n → ∞, a result which was anticipated already in [8]. Further experimental results and improved bounds were obtained in [4, 6, 10, 12, 13, 14, 15, 16, 23, 24].
In a series of papers [19, 20, 21] from 1985–1987, Marc Mézard and Giorgio Parisi gave strong
evidence for the conjecture that as n → ∞,
π2
E Cn →
.
6
The first proof of (1) was found by David Aldous in 2000 [1, 2].
1
RESEARCH SUPPORTED BY THE SWEDISH RESEARCH COUNCIL
261
(1)
262
Electronic Communications in Probability
In 1998, Parisi conjectured [25] that
1 1
1
E Cn = 1 + + + · · · + 2 .
4 9
n
(2)
This suggested a proof by induction on n. The hope of finding such a proof increased further when
Don Coppersmith and Gregory Sorkin [6] extended the conjecture (2) to general k, m and n. They
suggested that
X
1
,
(3)
E Ck,m,n =
(m − i)(n − j)
i, j≥0
i+ j 0, and all other edges are exponential of rate 1. This corresponds in the Buck-Chan-Robbins model to letting am+1 have weight λ, and all other vertices have
weight 1. The following lemma is a special case of Lemma 5 of [5], where the authors speculate
that “This result may be the reason that simple formulas exist...”. We believe that they were right.
Lemma 2.2. Condition on the event that am+1 does not participate in σ r . Then the probability that
it participates in σ r+1 is
λ
.
(5)
m−r +λ
Proof. Suppose without loss of generality that the vertices of A participating in σ r are a1 , . . . , a r .
Now form a “contraction” K ′ of the original graph K by identifying the vertices a r+1 , . . . , am+1 to a
vertex a′r+1 (so that in K ′ there are multiple edges from a′r+1 ).
We condition on the cost of the minimum edge between each pair of vertices in K ′ . This can easily
be visualized in the matrix setting. The matrix entries are divided into blocks consisting either of
a single matrix entry Mi, j for i ≤ r, or of the set of matrix entries M r+1, j , . . . , Mm+1, j , see Figure 1.
We know the minimum cost of the edges within each block, but not the location of the edge having
this minimum cost.
It follows from Lemma 2.1 that σ r+1 cannot contain two edges from a r+1 , . . . , am+1 . Therefore
σ r+1 is essentially determined by the minimum (r + 1)-assignment σ′r+1 in K ′ . Once we know
the edge from a′r+1 that belongs to σ′r+1 , we know that it corresponds to the unique edge from
{a r+1 , . . . , am+1 } that belongs to σ r+1 . It follows from the “memorylessness” of the exponential
distribution that the unique vertex of a r+1 , . . . , am+1 that participates in σ r+1 is distributed with
probabilities proportional to the rates of the edge costs. This gives probability equal to (5) for the
vertex am+1 .
263
264
Electronic Communications in Probability
Corollary 2.3. The probability that am+1 participates in σk is
1−
m
·
m−1
m+λ m−1+λ
m−k+1
=
m−k+1+λ
−1
λ −1
λ
1− 1+
... 1 +
=
m
m−k+1
1
1
1
λ + O(λ2 ),
+
+ ··· +
m m−1
m−k+1
...
as λ → 0.
Proof. This follows from Lemmas 2.1 and 2.2.
3 Proof of the Coppersmith-Sorkin formula
We show that the Coppersmith-Sorkin formula (3) can easily be deduced from Corollary 2.3. The
reason that this was overlooked for several years is probably that it seems that by letting λ → 0,
we eliminate the extra vertex am+1 and just get the original problem back.
We let X be the cost of the minimum k-assignment in the m by n graph {a1 , . . . , am } × {b1 , . . . , bn }
and let Y be the cost of the minimum (k − 1)-assignment in the m by n − 1 graph {a1 , . . . , am } ×
{b1 , . . . , bn−1 }. Clearly X and Y are essentially the same as Ck,m,n and Ck−1,m,n−1 respectively, but
in this model, X and Y are also coupled in a specific way.
We let w denote the cost of the edge (am+1 , bn ), and let I be the indicator variable for the event
that the cost of the cheapest k-assignment that contains this edge is smaller than the cost of the
cheapest k-assignment that does not use am+1 . In other words, I is the indicator variable for the
event that Y + w < X .
Lemma 3.1. In the limit λ → 0,
1
1
1
λ + O(λ2 ).
+
+ ··· +
E (I) =
mn (m − 1)n
(m − k + 1)n
(6)
Proof. It follows from Corollary 2.3 that the probability that (am+1 , bn ) participates in the minimum k-assignment is given by (6). If it does, then w < X − Y . Conversely, if w < X − Y and
no other edge from am+1 has cost smaller than X , then (am+1 , bn ) participates in the minimum
k-assignment, and when λ → 0, the probability that there are two distinct edges from am+1 of cost
smaller than X is of order O(λ2 ).
On the other hand, the fact that w is exponentially distributed of rate λ means that
E (I) = P(w < X − Y ) = E 1 − e−λ(X −Y ) = 1 − E e−λ(X −Y ) .
Hence E (I), regarded as a function of λ, is essentially the Laplace transform of X −Y . In particular
E (X − Y ) is the derivative of E (I) evaluated at λ = 0:
E (X − Y ) =
d
dλ
E (I) |λ=0 =
1
mn
+
This establishes (4) and thereby (3), (2) and (1).
1
(m − 1)n
+ ··· +
1
(m − k + 1)n
.
An easy proof of the ζ(2) limit
265
4 A bound on the variance
The Parisi formula (2) shows that as n → ∞, E(Cn ) converges to ζ(2) = π2 /6. To establish ζ(2)
as a “universal constant” for the assignment problem, it is also of interest to prove convergence in
probability. This can be done by showing that var(Cn ) → 0. The upper bound
(log n)4
var(Cn ) = O
n(log log n)2
was obtained by Michel Talagrand [26] in 1995 by an application of his isoperimetric inequality.
In [28] it was shown that
4ζ(2) − 4ζ(3)
.
(7)
var(Cn ) ∼
n
These proofs are both quite complicated, and our purpose here is to present a relatively simple
argument demonstrating that var(Cn ) = O(1/n).
We first establish a simple correlation inequality which is closely related to the Harris inequality
[11]. Let X 1 , . . . , X N be random variables (not necessarily independent), and let f and g be
two real valued functions of X 1 , . . . , X N . For 0 ≤ i ≤ N , let f i = E( f |X 1 , . . . , X i ), and similarly
g i = E(g|X 1 , . . . , X i ). In particular f0 = E( f ), f N = f , and similarly for g. The following lemma
requires that these and certain other expectations are well-defined. Let us simply assume that
regardless of X 1 , . . . , X i , all the conditional moments of f and g are finite, since this will clearly
hold in the application we have in mind.
Lemma 4.1. Suppose that for every i and every outcome of X 1 , . . . , X N ,
( f i+1 − f i )(g i+1 − g i ) ≥ 0.
(8)
Then f and g are positively correlated, in other words,
E( f g) ≥ E( f )E(g).
(9)
Proof. Equation (8) can be written
f i+1 g i+1 ≥ ( f i+1 − f i )g i + (g i+1 − g i ) f i + f i g i .
Notice that f i+1 − f i , although not in general independent of g i ,has zero expectation conditioning
on X 1 , . . . , X i and thereby on g i . It follows that E ( f i+1 − f i )g i = 0, and similarly for the second
term. We conclude that E( f i+1 g i+1 ) ≥ E( f i g i ), and by induction that
E( f g) = E( f N g N ) ≥ E( f0 g0 ) = f0 g0 = E( f )E(g).
The random graph model that we use is the same as in the previous section, but we modify the
concept of “assignment” by allowing an arbitrary number of edges from the special vertex am+1
(but still at most one edge from each other vertex). This is not essential for the argument, but
simplifies some details. Lemmas 2.1 and 2.2 as well as Corollary 2.3 are still valid in this setting.
We let C be the cost of the minimum k-assignment σk (with the modified definition), and we let
J be the indicator variable for the event that am+1 participates in σk .
266
Electronic Communications in Probability
Lemma 4.2.
E (C · J) ≤ E (C) · E (J) .
(10)
Proof. Let f = C, and let g = 1 − J be the indicator variable for the event that am+1 does not
participate in σk . As the notation indicates, we are going to design a random process X 1 , . . . , X N
such that Lemma 4.1 applies. This process is governed by the edge costs, and X 1 , . . . , X N will give
us successively more information about the edge costs, until σk and its cost are determined. A
generic step of the process is similar to the situation in the proof of Lemma 2.2.
We let M (r) be the matrix of “blocks” when σ r is known, that is, the r + 1 by n matrix of block
minima as in Figure 1. Moreover we let θ1 , . . . , θk be vertices in A such that for r ≤ k, σ r uses the
vertices θ1 , . . . , θ r .
When we apply Lemma 4.1, the sequence X 1 , . . . , X N is taken to be the sequence
M (0), θ1 , M (1), θ2 , M (2), . . . , θk .
Notice first that the cost f of the minimum k-assignment is determined by M (k − 1), and that
θ1 , . . . , θk determine g, that is, they determine whether or not am+1 participates in σk .
In order to apply the lemma, we have to verify that each time we get a new piece of information,
the conditional expectations of f and g change in the same direction, if they change. By the
argument in the proof of Lemma 2.2, we have
E g|M (0), θ1 , . . . , M (r − 1), θ r = E g|M (0), θ1 , . . . , M (r − 1), θ r , M (r)
=
m−r −1
m−k+1
m−r
·
···
,
m−r +λ m−r −1+λ
m−k+1+λ
unless vn+1 ∈ {θ1 , . . . , θ r }, but in that case it is already clear that g = 0. Therefore when we get to
know another row in the matrix, the conditional expectation of g does not change, which means
that for this case, the hypothesis of Lemma 4.1 holds.
The other case to consider is when we already know M (0), . . . , M (r) and θ1 , . . . , θ r , and are being
informed of θ r+1 . In this case the conditional expectations of f and g can obviously both change.
For g, there are only two possibilities. Either θ r+1 = am+1 , which means that g = 0, or θ r+1 6=
am+1 , which implies that the conditional expectation of g increases.
To verify the hypothesis of Lemma 4.1, it clearly suffices to assume that {θ1 , . . . , θ r } = {1, . . . , r},
and to show that if θ r+1 = am+1 , then the conditional expectation of f decreases. Since we know
M (r), we know to which “block” of the matrix the new edge belongs, that is, there is a j such that
we know that exactly one of the edges in the set E ′ = {(ai , b j ) : r + 1 ≤ i ≤ m + 1} will belong to
σ r+1 .
We now condition on the costs of all edges that are not in E ′ . Since we know M (r), we also know
the minimum edge cost, say α, in E ′ . We now observe that if the minimum cost edge in E ′ is
(am+1 , b j ), then no other edge in E ′ can participate in σk , because in an assignment, any edge in
E ′ can be replaced by (am+1 , b j ). It follows that the value of f given that Mm+1, j = α is the same
regardless of the costs of the other edges in E ′ . In particular it is the same as the value of f given
that all edges in E ′ have cost α, which is certainly not greater than the conditional expecation of
f given that some edge other than (am+1 , b j ) has the minimum cost α in E ′ .
It follows that Lemma 4.1 applies, and this completes the proof.
The inequality (10) allows us to establish an upper bound on var(Cn ) which is of the right order
of magnitude (it is easy to see that var(Cn ) ≥ 1/n, see [3]).
An easy proof of the ζ(2) limit
267
Theorem 4.3.
var(Cn ) <
π2
3n
.
Proof. We let X , Y , I and w be as in Section 3, with I being the indicator variable of the event
Y + w < X . Again C denotes the cost of σk , and J is the indicator variable for the event that am+1
participates in σk .
Obviously
E (C) ≤ E (X ) .
(11)
Again the probability that there are two distinct edges from am+1 of cost smaller than X is of order
O(λ2 ). Therefore
E (J) = nE (I) + O(λ2 ) = nλE (X − Y ) + O(λ2 ).
(12)
Similarly
E (C · J) = nE (I · (Y + w)) + O(λ2 ).
(13)
If we condition on X and Y , then
E (I · (Y + w)) =
Z
X −Y
λe−λt (Y + t) d t
0
= Y (X − Y )λ +
(X − Y )2
2
λ + O(λ2 ) =
1
2
X 2 − Y 2 λ + O(λ2 ).
(14)
If, in the inequality (10), we substitute the results of (11), (12), (13) and (14), then after dividing
by nλ we obtain
1 2
E X − Y 2 ≤ E (X )2 − E (X ) E (Y ) + O(λ).
2
After deleting the error term, this can be rearranged as
var(X ) − var(Y ) ≤ (E (X ) − E (Y ))2 .
But we already know that E (X ) − E (Y ) is given by (4). Therefore it follows inductively that
var(Cn ) ≤
n
X
1 1
i=1
i2
n
+ ··· +
1
n−i+1
2
≤
n
X
1
i=1
i2
log(n + 1/2) − log(n + 1/2 − i)
2
.
If we replace the sum over i by an integral with respect to a continuous variable, then the integrand
is convex, and
var(Cn ) ≤
Z
n+1/2
0
log(n + 1/2) − log(n + 1/2 − x)
x2
=
2
dx
1
n + 1/2
Z
1
0
log(1 − x)2
x2
dx =
2ζ(2)
n + 1/2
<
π2
3n
.
268
Electronic Communications in Probability
References
[1] Aldous, D., Asymptotics in the random assignment problem, Probab. Theory Relat. Fields, 93
(1992) 507–534. MR1183889
[2] Aldous, D., The ζ(2) limit in the random assignment problem, Random Structures Algorithms
18 (2001), no 4. 381–418. MR1839499
[3] Alm, S. E. and Sorkin, G. B, Exact expectations and distributions in the random assignment
problem, Combin. Probab. Comput. 11 (2002), no. 3, 217–248. MR1909500
[4] Brunetti, R., Krauth, W., Mézard, M. and Parisi, G., Extensive Numerical Simulation of
Weighted Matchings: Total Length and Distribuion of Links in the Optimal Solution, Europhys.
Lett. 14 4 (1991), 295–301.
[5] Buck, M. W., Chan, C. S. and Robbins, D. P., On the expected value of the minimum assignment,
Random Structures & Algorithms 21 (2002), no. 1, 33–58 (also arXiv:math.CO/0004175).
MR1913077
[6] Coppersmith, D. and Sorkin, G. B., Constructive Bounds and Exact Expectations For the
Random Assignment Problem, Random Structures & Algorithms 15 (1999), 133–144.
MR1704340
[7] Coppersmith, D. and Sorkin, G. B., On the expected incremental cost of a minimum assignment.
In Contemporary Mathematics (B. Bollobás, ed.), Vol. 10 of Bolyai Society Mathematical
Studies, Springer. MR1919573
[8] Donath, W. E., Algorithm and average-value bounds for assignment problems, IBM J. Res. Dev.
13 (1969), 380–386.
[9] Dotsenko, V., Exact solution of the random bipartite matching model, J. Phys. A, 33 (2000),
2015–2030, arXiv, cond-mat/9911477. MR1748747
[10] Goemans, M. X., and Kodialam, M. S., A lower bound on the expected cost of an optimal
assignment, Math. Oper. Res., 18 (1993), 267–274. MR1250118
[11] Harris, T. E., Lower bound for the critical probability in a certain percolation process, Proc.
Cambridge Phil. Soc. 56 (1960), 13–20. MR0115221
[12] Houdayer, J., Bouter de Monvel, J. H. and Martin, O. C. Comparing mean field and Euclidean
matching problems, Eur. Phys. J. B. 6 (1998), 383–393. MR1665617
[13] Karp, R. M., An upper bound on the expected cost of an optimal assignment, In Discrete Algorithms and Complexity: Proceedings of the Japan-U.S. Joint Seminar, Academic Press, 1987,
1–4. MR0910922
[14] Karp, R., Rinnooy Kan, A. H. G. and Vohra, R. V., Average case analysis of a heuristic for the
assignment problem, Math. Oper. Res. 19 (1994), 513–522. MR1288884
[15] Lazarus, A. J., The assignment problem with uniform (0,1) cost matrix, Master’s thesis, Department of Mathematics, Princeton University, Princeton NJ, 1979.
[16] Lazarus, A. J., Certain expected values in the random assignment problem, Oper. Res. Lett., 14
(1993), 207–214. MR1250142
An easy proof of the ζ(2) limit
[17] Linusson, S. and Wästlund, J., A generalization of the random assignment problem,
arXiv:math.CO/0006146.
[18] Linusson, S. and Wästlund, J., A proof of Parisi’s conjecture on the random assignment problem,
Probab. Theory Relat. Fields 128 (2004), 419–440. MR2036492
[19] Mézard, M. and Parisi, G., Replicas and optimization, J. Phys. Lett. 46(1985), 771–778.
[20] Mézard, M. and Parisi, G., Mean-field equations for the matching and the travelling salesman
problems, Europhys. Lett. 2 (1986) 913–918.
[21] Mézard, M., and Parisi, G., On the solution of the random link matching problems, J. Physique
Lettres 48 (1987),1451–1459.
[22] Nair, C., Prabhakar, B. and Sharma, M., Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures, Random Structures and Algorithms 27 No. 4 (2006), 413–444.
MR2178256
[23] Olin, B., Asymptotic properties of the random assignment problem, Ph.D. thesis, Kungl Tekniska
Högskolan, Stockholm, Sweden, (1992).
[24] Pardalos, P. M. and Ramakrishnan, K. G., On the expected optimal value of random assignment
problems: Experimental results and open questions, Comput. Optim. Appl. 2 (1993), 261–271.
MR1258206
[25] Parisi, G., A conjecture on random bipartite matching,
arXiv:cond-mat/9801176, 1998.
[26] Talagrand, M., Concentration of measure and isoperimetric inequalities in product spaces, Inst.
Hautes Études Sci. Publ. Math. 81 (1995), 73–205. MR1361756
[27] Walkup, D. W., On the expected value of a random assignment problem, SIAM J. Comput., 8
(1979), 440–442. MR0539262
[28] Wästlund, J., The variance and higher moments in the random assignment problem, Linköping
Studies in Mathematics no 8 (2005), http://www.ep.liu.se/ea/lsm/2005/008/.
269
ELECTRONIC
COMMUNICATIONS
in PROBABILITY
AN EASY PROOF OF THE ζ(2) LIMIT IN THE RANDOM ASSIGNMENT PROBLEM
JOHAN WÄSTLUND 1
Department of Mathematical Sciences,
Chalmers University of Technology,
S-412 96 Gothenburg, Sweden
email: [email protected]
Submitted April 4, 2008, accepted in final form December 3, 2008
AMS 2000 Subject classification: 60C05, 90C27, 90C35
Keywords: minimum matching, graph, exponential
Abstract
The edges of the complete bipartite graph Kn,n are given independent exponentially distributed
costs. Let Cn be the minimum total cost of a perfect matching. It was conjectured by M. Mézard
and G. Parisi in 1985, and proved by D. Aldous in 2000, that Cn converges in probability to π2 /6.
We give a short proof of this fact, consisting of a proof of the exact formula 1+1/4+1/9+· · ·+1/n2
for the expectation of Cn , and a O(1/n) bound on the variance.
1 Introduction
We consider the following random model of the assignment problem: The edges of an m by n
complete bipartite graph are assigned independent exponentially distributed costs. A k-assignment
is a set of k edges of which no two have a vertex in common. The cost of an assignment is the
sum of the costs of its edges. Equivalently, if the costs are represented by an m by n matrix, a
k-assignment is a set of k matrix entries, no two in the same row or column. We let Ck,m,n denote
the minimum cost of a k-assignment. We are primarily interested in the case k = m = n, where
we write Cn = Cn,n,n .
The distribution
of Cn has been investigated for several decades. In 1979, D. Walkup [27] showed
that E Cn is bounded as n → ∞, a result which was anticipated already in [8]. Further experimental results and improved bounds were obtained in [4, 6, 10, 12, 13, 14, 15, 16, 23, 24].
In a series of papers [19, 20, 21] from 1985–1987, Marc Mézard and Giorgio Parisi gave strong
evidence for the conjecture that as n → ∞,
π2
E Cn →
.
6
The first proof of (1) was found by David Aldous in 2000 [1, 2].
1
RESEARCH SUPPORTED BY THE SWEDISH RESEARCH COUNCIL
261
(1)
262
Electronic Communications in Probability
In 1998, Parisi conjectured [25] that
1 1
1
E Cn = 1 + + + · · · + 2 .
4 9
n
(2)
This suggested a proof by induction on n. The hope of finding such a proof increased further when
Don Coppersmith and Gregory Sorkin [6] extended the conjecture (2) to general k, m and n. They
suggested that
X
1
,
(3)
E Ck,m,n =
(m − i)(n − j)
i, j≥0
i+ j 0, and all other edges are exponential of rate 1. This corresponds in the Buck-Chan-Robbins model to letting am+1 have weight λ, and all other vertices have
weight 1. The following lemma is a special case of Lemma 5 of [5], where the authors speculate
that “This result may be the reason that simple formulas exist...”. We believe that they were right.
Lemma 2.2. Condition on the event that am+1 does not participate in σ r . Then the probability that
it participates in σ r+1 is
λ
.
(5)
m−r +λ
Proof. Suppose without loss of generality that the vertices of A participating in σ r are a1 , . . . , a r .
Now form a “contraction” K ′ of the original graph K by identifying the vertices a r+1 , . . . , am+1 to a
vertex a′r+1 (so that in K ′ there are multiple edges from a′r+1 ).
We condition on the cost of the minimum edge between each pair of vertices in K ′ . This can easily
be visualized in the matrix setting. The matrix entries are divided into blocks consisting either of
a single matrix entry Mi, j for i ≤ r, or of the set of matrix entries M r+1, j , . . . , Mm+1, j , see Figure 1.
We know the minimum cost of the edges within each block, but not the location of the edge having
this minimum cost.
It follows from Lemma 2.1 that σ r+1 cannot contain two edges from a r+1 , . . . , am+1 . Therefore
σ r+1 is essentially determined by the minimum (r + 1)-assignment σ′r+1 in K ′ . Once we know
the edge from a′r+1 that belongs to σ′r+1 , we know that it corresponds to the unique edge from
{a r+1 , . . . , am+1 } that belongs to σ r+1 . It follows from the “memorylessness” of the exponential
distribution that the unique vertex of a r+1 , . . . , am+1 that participates in σ r+1 is distributed with
probabilities proportional to the rates of the edge costs. This gives probability equal to (5) for the
vertex am+1 .
263
264
Electronic Communications in Probability
Corollary 2.3. The probability that am+1 participates in σk is
1−
m
·
m−1
m+λ m−1+λ
m−k+1
=
m−k+1+λ
−1
λ −1
λ
1− 1+
... 1 +
=
m
m−k+1
1
1
1
λ + O(λ2 ),
+
+ ··· +
m m−1
m−k+1
...
as λ → 0.
Proof. This follows from Lemmas 2.1 and 2.2.
3 Proof of the Coppersmith-Sorkin formula
We show that the Coppersmith-Sorkin formula (3) can easily be deduced from Corollary 2.3. The
reason that this was overlooked for several years is probably that it seems that by letting λ → 0,
we eliminate the extra vertex am+1 and just get the original problem back.
We let X be the cost of the minimum k-assignment in the m by n graph {a1 , . . . , am } × {b1 , . . . , bn }
and let Y be the cost of the minimum (k − 1)-assignment in the m by n − 1 graph {a1 , . . . , am } ×
{b1 , . . . , bn−1 }. Clearly X and Y are essentially the same as Ck,m,n and Ck−1,m,n−1 respectively, but
in this model, X and Y are also coupled in a specific way.
We let w denote the cost of the edge (am+1 , bn ), and let I be the indicator variable for the event
that the cost of the cheapest k-assignment that contains this edge is smaller than the cost of the
cheapest k-assignment that does not use am+1 . In other words, I is the indicator variable for the
event that Y + w < X .
Lemma 3.1. In the limit λ → 0,
1
1
1
λ + O(λ2 ).
+
+ ··· +
E (I) =
mn (m − 1)n
(m − k + 1)n
(6)
Proof. It follows from Corollary 2.3 that the probability that (am+1 , bn ) participates in the minimum k-assignment is given by (6). If it does, then w < X − Y . Conversely, if w < X − Y and
no other edge from am+1 has cost smaller than X , then (am+1 , bn ) participates in the minimum
k-assignment, and when λ → 0, the probability that there are two distinct edges from am+1 of cost
smaller than X is of order O(λ2 ).
On the other hand, the fact that w is exponentially distributed of rate λ means that
E (I) = P(w < X − Y ) = E 1 − e−λ(X −Y ) = 1 − E e−λ(X −Y ) .
Hence E (I), regarded as a function of λ, is essentially the Laplace transform of X −Y . In particular
E (X − Y ) is the derivative of E (I) evaluated at λ = 0:
E (X − Y ) =
d
dλ
E (I) |λ=0 =
1
mn
+
This establishes (4) and thereby (3), (2) and (1).
1
(m − 1)n
+ ··· +
1
(m − k + 1)n
.
An easy proof of the ζ(2) limit
265
4 A bound on the variance
The Parisi formula (2) shows that as n → ∞, E(Cn ) converges to ζ(2) = π2 /6. To establish ζ(2)
as a “universal constant” for the assignment problem, it is also of interest to prove convergence in
probability. This can be done by showing that var(Cn ) → 0. The upper bound
(log n)4
var(Cn ) = O
n(log log n)2
was obtained by Michel Talagrand [26] in 1995 by an application of his isoperimetric inequality.
In [28] it was shown that
4ζ(2) − 4ζ(3)
.
(7)
var(Cn ) ∼
n
These proofs are both quite complicated, and our purpose here is to present a relatively simple
argument demonstrating that var(Cn ) = O(1/n).
We first establish a simple correlation inequality which is closely related to the Harris inequality
[11]. Let X 1 , . . . , X N be random variables (not necessarily independent), and let f and g be
two real valued functions of X 1 , . . . , X N . For 0 ≤ i ≤ N , let f i = E( f |X 1 , . . . , X i ), and similarly
g i = E(g|X 1 , . . . , X i ). In particular f0 = E( f ), f N = f , and similarly for g. The following lemma
requires that these and certain other expectations are well-defined. Let us simply assume that
regardless of X 1 , . . . , X i , all the conditional moments of f and g are finite, since this will clearly
hold in the application we have in mind.
Lemma 4.1. Suppose that for every i and every outcome of X 1 , . . . , X N ,
( f i+1 − f i )(g i+1 − g i ) ≥ 0.
(8)
Then f and g are positively correlated, in other words,
E( f g) ≥ E( f )E(g).
(9)
Proof. Equation (8) can be written
f i+1 g i+1 ≥ ( f i+1 − f i )g i + (g i+1 − g i ) f i + f i g i .
Notice that f i+1 − f i , although not in general independent of g i ,has zero expectation conditioning
on X 1 , . . . , X i and thereby on g i . It follows that E ( f i+1 − f i )g i = 0, and similarly for the second
term. We conclude that E( f i+1 g i+1 ) ≥ E( f i g i ), and by induction that
E( f g) = E( f N g N ) ≥ E( f0 g0 ) = f0 g0 = E( f )E(g).
The random graph model that we use is the same as in the previous section, but we modify the
concept of “assignment” by allowing an arbitrary number of edges from the special vertex am+1
(but still at most one edge from each other vertex). This is not essential for the argument, but
simplifies some details. Lemmas 2.1 and 2.2 as well as Corollary 2.3 are still valid in this setting.
We let C be the cost of the minimum k-assignment σk (with the modified definition), and we let
J be the indicator variable for the event that am+1 participates in σk .
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Electronic Communications in Probability
Lemma 4.2.
E (C · J) ≤ E (C) · E (J) .
(10)
Proof. Let f = C, and let g = 1 − J be the indicator variable for the event that am+1 does not
participate in σk . As the notation indicates, we are going to design a random process X 1 , . . . , X N
such that Lemma 4.1 applies. This process is governed by the edge costs, and X 1 , . . . , X N will give
us successively more information about the edge costs, until σk and its cost are determined. A
generic step of the process is similar to the situation in the proof of Lemma 2.2.
We let M (r) be the matrix of “blocks” when σ r is known, that is, the r + 1 by n matrix of block
minima as in Figure 1. Moreover we let θ1 , . . . , θk be vertices in A such that for r ≤ k, σ r uses the
vertices θ1 , . . . , θ r .
When we apply Lemma 4.1, the sequence X 1 , . . . , X N is taken to be the sequence
M (0), θ1 , M (1), θ2 , M (2), . . . , θk .
Notice first that the cost f of the minimum k-assignment is determined by M (k − 1), and that
θ1 , . . . , θk determine g, that is, they determine whether or not am+1 participates in σk .
In order to apply the lemma, we have to verify that each time we get a new piece of information,
the conditional expectations of f and g change in the same direction, if they change. By the
argument in the proof of Lemma 2.2, we have
E g|M (0), θ1 , . . . , M (r − 1), θ r = E g|M (0), θ1 , . . . , M (r − 1), θ r , M (r)
=
m−r −1
m−k+1
m−r
·
···
,
m−r +λ m−r −1+λ
m−k+1+λ
unless vn+1 ∈ {θ1 , . . . , θ r }, but in that case it is already clear that g = 0. Therefore when we get to
know another row in the matrix, the conditional expectation of g does not change, which means
that for this case, the hypothesis of Lemma 4.1 holds.
The other case to consider is when we already know M (0), . . . , M (r) and θ1 , . . . , θ r , and are being
informed of θ r+1 . In this case the conditional expectations of f and g can obviously both change.
For g, there are only two possibilities. Either θ r+1 = am+1 , which means that g = 0, or θ r+1 6=
am+1 , which implies that the conditional expectation of g increases.
To verify the hypothesis of Lemma 4.1, it clearly suffices to assume that {θ1 , . . . , θ r } = {1, . . . , r},
and to show that if θ r+1 = am+1 , then the conditional expectation of f decreases. Since we know
M (r), we know to which “block” of the matrix the new edge belongs, that is, there is a j such that
we know that exactly one of the edges in the set E ′ = {(ai , b j ) : r + 1 ≤ i ≤ m + 1} will belong to
σ r+1 .
We now condition on the costs of all edges that are not in E ′ . Since we know M (r), we also know
the minimum edge cost, say α, in E ′ . We now observe that if the minimum cost edge in E ′ is
(am+1 , b j ), then no other edge in E ′ can participate in σk , because in an assignment, any edge in
E ′ can be replaced by (am+1 , b j ). It follows that the value of f given that Mm+1, j = α is the same
regardless of the costs of the other edges in E ′ . In particular it is the same as the value of f given
that all edges in E ′ have cost α, which is certainly not greater than the conditional expecation of
f given that some edge other than (am+1 , b j ) has the minimum cost α in E ′ .
It follows that Lemma 4.1 applies, and this completes the proof.
The inequality (10) allows us to establish an upper bound on var(Cn ) which is of the right order
of magnitude (it is easy to see that var(Cn ) ≥ 1/n, see [3]).
An easy proof of the ζ(2) limit
267
Theorem 4.3.
var(Cn ) <
π2
3n
.
Proof. We let X , Y , I and w be as in Section 3, with I being the indicator variable of the event
Y + w < X . Again C denotes the cost of σk , and J is the indicator variable for the event that am+1
participates in σk .
Obviously
E (C) ≤ E (X ) .
(11)
Again the probability that there are two distinct edges from am+1 of cost smaller than X is of order
O(λ2 ). Therefore
E (J) = nE (I) + O(λ2 ) = nλE (X − Y ) + O(λ2 ).
(12)
Similarly
E (C · J) = nE (I · (Y + w)) + O(λ2 ).
(13)
If we condition on X and Y , then
E (I · (Y + w)) =
Z
X −Y
λe−λt (Y + t) d t
0
= Y (X − Y )λ +
(X − Y )2
2
λ + O(λ2 ) =
1
2
X 2 − Y 2 λ + O(λ2 ).
(14)
If, in the inequality (10), we substitute the results of (11), (12), (13) and (14), then after dividing
by nλ we obtain
1 2
E X − Y 2 ≤ E (X )2 − E (X ) E (Y ) + O(λ).
2
After deleting the error term, this can be rearranged as
var(X ) − var(Y ) ≤ (E (X ) − E (Y ))2 .
But we already know that E (X ) − E (Y ) is given by (4). Therefore it follows inductively that
var(Cn ) ≤
n
X
1 1
i=1
i2
n
+ ··· +
1
n−i+1
2
≤
n
X
1
i=1
i2
log(n + 1/2) − log(n + 1/2 − i)
2
.
If we replace the sum over i by an integral with respect to a continuous variable, then the integrand
is convex, and
var(Cn ) ≤
Z
n+1/2
0
log(n + 1/2) − log(n + 1/2 − x)
x2
=
2
dx
1
n + 1/2
Z
1
0
log(1 − x)2
x2
dx =
2ζ(2)
n + 1/2
<
π2
3n
.
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Electronic Communications in Probability
References
[1] Aldous, D., Asymptotics in the random assignment problem, Probab. Theory Relat. Fields, 93
(1992) 507–534. MR1183889
[2] Aldous, D., The ζ(2) limit in the random assignment problem, Random Structures Algorithms
18 (2001), no 4. 381–418. MR1839499
[3] Alm, S. E. and Sorkin, G. B, Exact expectations and distributions in the random assignment
problem, Combin. Probab. Comput. 11 (2002), no. 3, 217–248. MR1909500
[4] Brunetti, R., Krauth, W., Mézard, M. and Parisi, G., Extensive Numerical Simulation of
Weighted Matchings: Total Length and Distribuion of Links in the Optimal Solution, Europhys.
Lett. 14 4 (1991), 295–301.
[5] Buck, M. W., Chan, C. S. and Robbins, D. P., On the expected value of the minimum assignment,
Random Structures & Algorithms 21 (2002), no. 1, 33–58 (also arXiv:math.CO/0004175).
MR1913077
[6] Coppersmith, D. and Sorkin, G. B., Constructive Bounds and Exact Expectations For the
Random Assignment Problem, Random Structures & Algorithms 15 (1999), 133–144.
MR1704340
[7] Coppersmith, D. and Sorkin, G. B., On the expected incremental cost of a minimum assignment.
In Contemporary Mathematics (B. Bollobás, ed.), Vol. 10 of Bolyai Society Mathematical
Studies, Springer. MR1919573
[8] Donath, W. E., Algorithm and average-value bounds for assignment problems, IBM J. Res. Dev.
13 (1969), 380–386.
[9] Dotsenko, V., Exact solution of the random bipartite matching model, J. Phys. A, 33 (2000),
2015–2030, arXiv, cond-mat/9911477. MR1748747
[10] Goemans, M. X., and Kodialam, M. S., A lower bound on the expected cost of an optimal
assignment, Math. Oper. Res., 18 (1993), 267–274. MR1250118
[11] Harris, T. E., Lower bound for the critical probability in a certain percolation process, Proc.
Cambridge Phil. Soc. 56 (1960), 13–20. MR0115221
[12] Houdayer, J., Bouter de Monvel, J. H. and Martin, O. C. Comparing mean field and Euclidean
matching problems, Eur. Phys. J. B. 6 (1998), 383–393. MR1665617
[13] Karp, R. M., An upper bound on the expected cost of an optimal assignment, In Discrete Algorithms and Complexity: Proceedings of the Japan-U.S. Joint Seminar, Academic Press, 1987,
1–4. MR0910922
[14] Karp, R., Rinnooy Kan, A. H. G. and Vohra, R. V., Average case analysis of a heuristic for the
assignment problem, Math. Oper. Res. 19 (1994), 513–522. MR1288884
[15] Lazarus, A. J., The assignment problem with uniform (0,1) cost matrix, Master’s thesis, Department of Mathematics, Princeton University, Princeton NJ, 1979.
[16] Lazarus, A. J., Certain expected values in the random assignment problem, Oper. Res. Lett., 14
(1993), 207–214. MR1250142
An easy proof of the ζ(2) limit
[17] Linusson, S. and Wästlund, J., A generalization of the random assignment problem,
arXiv:math.CO/0006146.
[18] Linusson, S. and Wästlund, J., A proof of Parisi’s conjecture on the random assignment problem,
Probab. Theory Relat. Fields 128 (2004), 419–440. MR2036492
[19] Mézard, M. and Parisi, G., Replicas and optimization, J. Phys. Lett. 46(1985), 771–778.
[20] Mézard, M. and Parisi, G., Mean-field equations for the matching and the travelling salesman
problems, Europhys. Lett. 2 (1986) 913–918.
[21] Mézard, M., and Parisi, G., On the solution of the random link matching problems, J. Physique
Lettres 48 (1987),1451–1459.
[22] Nair, C., Prabhakar, B. and Sharma, M., Proofs of the Parisi and Coppersmith-Sorkin random assignment conjectures, Random Structures and Algorithms 27 No. 4 (2006), 413–444.
MR2178256
[23] Olin, B., Asymptotic properties of the random assignment problem, Ph.D. thesis, Kungl Tekniska
Högskolan, Stockholm, Sweden, (1992).
[24] Pardalos, P. M. and Ramakrishnan, K. G., On the expected optimal value of random assignment
problems: Experimental results and open questions, Comput. Optim. Appl. 2 (1993), 261–271.
MR1258206
[25] Parisi, G., A conjecture on random bipartite matching,
arXiv:cond-mat/9801176, 1998.
[26] Talagrand, M., Concentration of measure and isoperimetric inequalities in product spaces, Inst.
Hautes Études Sci. Publ. Math. 81 (1995), 73–205. MR1361756
[27] Walkup, D. W., On the expected value of a random assignment problem, SIAM J. Comput., 8
(1979), 440–442. MR0539262
[28] Wästlund, J., The variance and higher moments in the random assignment problem, Linköping
Studies in Mathematics no 8 (2005), http://www.ep.liu.se/ea/lsm/2005/008/.
269